RI: Small:Differential Ray Geometry for Surface Reconstruction and Modeling

本项目重点发展微分射线几何的理论基础。PI首先推导出一个全面的射线几何框架，包括射线畸变、射线焦散、射线曲率理论以及射线微分算子。这个新框架广泛适用于现实世界的问题。在计算机视觉方面，PI探索了鲁棒和有效的方案，从扭曲或焦散模式推断光线结构，然后从光线微分属性恢复表面几何形状。这导致了一类新的镜面（反射和折射）表面重建算法。在计算机图形方面，PI采用了一种新颖的法向射线表示，将光滑的3D表面转换为2D射线流形，从而可以直接从法向射线几何中导出表面微分属性。PI进一步开发了新的细分、重新网格划分和网格简化方案，以生成与底层法线结构一致的表面。这项研究通过提供相机、光源和表面的微分射线几何模型，使许多计算机视觉和图形应用受益。它也有利于飞机、汽车和许多其他需要高阶形状一致性的行业的形状设计。该项目通过开设新的微分几何课程和研讨会，以及让妇女和代表性不足的学生参与数学和计算机科学研究，为教育作出贡献。PI进一步寻求通过该项目与数学和物理、光学工程和机械工程领域建立牢固的联系。

This project focuses on developing the theoretical foundation of differential ray geometry. The PI first derives a comprehensive ray geometry framework including the ray-distortion, ray-caustics, and ray-curvature theories as well as ray differential operators. This new framework is widely applicable to real-world problems. On the computer vision front, the PI explores robust and efficient schemes to infer ray structures from distortions or from caustics patterns and then recover surface geometry from ray differential attributes. This leads to a new class of specular (reflective and refractive) surface reconstruction algorithms. On the computer graphics front, the PI employs a novel normal-ray representation that converts a smooth 3D surface into a 2D ray manifold so that surface differential attributes can be directly derived from normal-ray geometry. The PI further develops new subdivision, re-meshing, and mesh simplification schemes for generating surfaces consistent with the underlying normal ray structures. This research benefits many computer vision and graphics applications by providing a differential ray geometry model for cameras, light sources, and surfaces. It also benefits shape designs in aircraft, automobile, and many other industries, where higher-order shape consistencies are required. This project contributes to education through the development of new differential geometry courses and seminars and by involving women and under-represented students in mathematical and computer science research. The PI further seeks to build strong connections with the fields of mathematics and physics, optical engineering, and mechanical engineering through the project.